Euler Singularities I: Boundary Blow-Up for Smooth Exact-Odd Axisymmetric Euler with Swirl

Abstract

We construct smooth axisymmetric-with-swirl initial data in a periodic cylinder for which the three-dimensional incompressible Euler evolution develops a finite-time boundary singularity. The construction is carried out in the dynamically invariant exact-odd class \[ \Gamma(r,-z,t)=-\Gamma(r,z,t), \qquad G(r,-z,t)=-G(r,z,t), \] where \(\Gamma=r u^\theta\) and \(G=\omega^\theta/r\). At the side-wall point \((r,z)=(1,0)\), exact oddness gives the pointwise identities \[ \partial_t\partial_zG(1,0,t) = \sigma(t)\partial_zG(1,0,t) +2\bigl(\partial_z\Gamma(1,0,t)\bigr)^2, \qquad \partial_t\partial_z\Gamma(1,0,t) = \sigma(t)\partial_z\Gamma(1,0,t), \] with \(\sigma(t)=-\partial_z u^z(1,0,t)\). The proof is based on a side-wall Dirichlet parametrix for the five-dimensional lifted recovery equation \(-\Delta_5\phi=G\). Near the wall, the effective compression kernel has leading term \[ K_0(x,y)=C_0\frac{xy}{(x^2+y^2)^2}, \qquad C_0>0, \] with controlled remainders, parity-based shear cancellation, and strain-variation bounds on narrow diagonal cones. These estimates are combined with an over-compressed dyadic angular cluster functional. The cluster functional absorbs same-scale angular fragmentation, growing dyadic windows, dynamically separated far tails, and fixed-distance exterior fields into an integrably small affine Campanato defect. The resulting invariant cluster contains a uniformly coherent component with amplitudes \(A_*(t)\) and \(B_*(t)\) satisfying the Dini comparison system \[ D^+A_*(t)\ge cB_*(t)^2, \qquad D^+B_*(t)\ge cA_*(t)B_*(t). \]